Spahn sheaf on a sheaf (Rev #20, changes)

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Motivation

Let X∈HX\in H be a space (an object of a category HH of spaces), let Sh(X)Sh(X) be the category of sheaves on the frame of opens on XX, let (H/X) et(H/X)^{et} denote the wide subcategory of H/XH/X with only étale morphisms. Then there is an adjoint equivalence

(L⊣Γ):(H/X) et→ΓSh(X)(L\dashv \Gamma):(H/X)^{et}\stackrel{\Gamma}{\to}Sh(X)

where

  • Γ\Gamma sends an étale morphism f:U→Xf:U\to X to the sheaf of local sections of ff.

  • LL sends a sheaf on XX to its espace étale.

Très petit topos

We wish to clarify in which sense also the (∞,1)(\infty,1)- topos (H/X) fet(H/X)^{fet} can be regarded as an (∞,1)(\infty,1)-sheaftopos on XX. One formulation of this is to show that ((H/X) fet,O (H/X) fet)((H/X)^{fet},O_{(H/X)^{fet}}) is a locally representable structured (∞,1)(\infty,1)-topos - and that the representation is exhibited by formally étale morphisms.

We assume that fetfet is the admissible class defined by an infinitesimal modality □\Box on HH.

Definition (universal GG-structure, classifying topos)

(1) A GG-structure OO on an (∞,1)(\infty,1)-topos is called universal if for every (∞,1)(\infty,1)-topos XX composition with OO induces an equivalence of (∞,1)(\infty,1)-categories

Fun *(K,T)→Str G(T)Fun^*(K,T)\to Str_G(T)

where Fun *(K,T)Fun^*(K,T) denotes the geometric morphisms ff with inverse image f *:K→Tf^*:K\to T.

(2) In this case we say OO exhibits KK as classifying (∞,1)(\infty,1)-topos for GG-structures on XX.

Remark

HH, H/XH/X, and (H/X) fet(H/X)^fet are (H/X) fet(H/X)^{fet}-structured (∞,1)(\infty,1)-toposes.

Proof

The classifying topos for (G/X) fet(G/X)^{fet}-structures is HH and the (∞,1)(\infty,1)-toposes in question are linked with HH by geometric morphisms. We obtain the required structures as the image of

Fun *(H,H/X)→Str (H/X) fet(H/X)Fun^*(H,H/X)\to Str_{(H/X)^{fet}}(H/X)

respectively for HH and (H/X) fet(H/X)^{fet} in place of “H/X”.

Local representability of the très petit topos

Definition (pro objects)

Let CC be an (∞,1)(\infty,1)-category. We have Ind(C op)≃Pro(C) opInd(C^{op})\simeq Pro(C)^{op}. A pro object in CC is a formal limit of a cofiltered diagram in CC. A cofiltered diagram is defined to be a finite diagram FF having a cone (i.e. a family of natural transformation κ c→F\kappa_c\to F for all c∈Cc\in C, where κ c\kappa_c denotes the constant functor having value cc). So we have

Pro(C)={F:D→C|Disfinite,cofiltered}Pro(C)=\{F:D\to C | D\,is\,finite,\,cofiltered\}

and the hom sets are

Pro(C)(F,G)=lim e∈Ecolim d∈DC(F(d),G(e))Pro(C)(F,G)=lim_{e\in E}colim_{d\in D}C(F(d),G(e))

We have (more or less) synonyms:

  • pro object, cofiltered, having a cone

  • ind object, filtered, having a cocone

Digression (DAG (Remark V, 2.2.7) Prop.2.3.7)

(1) Let A morphisms f H:(X,O G,X)→(Y,O G,Y) f:(X,O_{G,X})\to H (Y,O_{G,Y}) is be called a étale geometry. if Then (1a) every the admissible underlying geometric morphism of(U∞→,X1) (\infty,1) U\to X -toposes is inétalePro(H)Pro(H) and arises (1b) as the a induced pullback mapf *:X→Yf^*:X\to Y is an equivalence in Str G(𝔘)Str_G(\mathfrak{U})

(2) Condition (1b) is equivalent to the requirement that ff is pp-cocartesian for p:LTop(G)→LTopp:LTop(G)\to LTop the projection.

U → j(U ′) ↓ ↓ X → j(X ′)\array{ U&\to&j(U^\prime) \\ \downarrow&&\downarrow \\ X&\to&j(X^\prime) }

(3) in Being an étale geometric morphism of structuredPro(∞H,1) (\infty,1) Pro(H) -toposes . is a local property:

If there is an effective epimorphism ∐ iU i→* X\coprod_i U_i\to *_X to the terminal object of XX, and f:(X,O G,X)→(Y,O G,Y)f:(X,O_{G,X})\to (Y,O_{G,Y}) in LTop(G) opLTop(G)^{op} a morphism such that
f |U i:((X/U i,(O G,X) |U i)→(Y,O G,Y)f_{|U_i}:((X/U_i,(O_{G,X})_{|U_i})\to (Y,O_{G,Y})

is étale, then ff is étale.
Definition Digression (Restriction (DAG 2.3.3, V, DAG Prop.2.3.7) chapter 2.3)

Let (1) A morphismsf:(X,O G,X)→(Y,O G,Y) (X,O_{G,X}) f:(X,O_{G,X})\to (Y,O_{G,Y}) be is a called structured étale if (1a) the underlying geometric morphism of(∞,1)(\infty,1) -topos, -toposes let isU∈XU\in Xétale be and an (1b) object. the induced mapf *:X→Yf^*:X\to Y is an equivalence in Str G(𝔘)Str_G(\mathfrak{U})

(1) (2) The Condition restriction (1b) of is equivalent to the requirement that X f X f to is U p U p -cocartesian is for defined to be the slice X p / : U LTop(G)→LTop X/U p:LTop(G)\to LTop . the projection.

(2) (3) The Being restriction an étale geometric morphism of structured(O G,X∞) |U,1) (O_{G,X})_{| (\infty,1) U} -toposes of is a local property:O G,XO_{G,X} to UU is defined to be composite

G→O G,XX→p *X/UG\stackrel{O_{G,X}}{\to}X\stackrel{p^*}{\to}X/U

If there is an effective epimorphism ∐ iU i→* X\coprod_i U_i\to *_X to the terminal object of XX, and f:(X,O G,X)→(Y,O G,Y)f:(X,O_{G,X})\to (Y,O_{G,Y}) in LTop(G) opLTop(G)^{op} a morphism such that

where p *p^* is base change along p:U→*p:U\to *.

f |U i:((X/U i,(O G,X) |U i)→(Y,O G,Y)f_{|U_i}:((X/U_i,(O_{G,X})_{|U_i})\to (Y,O_{G,Y})

is étale, then ff is étale.
Definition ( (Restriction 2.3.3, DAG chapter 2.3)LTopLTop, LTop(G)LTop(G))

(1) LetLTop(X,O G,X) LTop (X,O_{G,X}) is defined to be the a structured(∞,1)(\infty,1) -category -topos, which let objects are(U∞∈,X1) (\infty,1) U\in X -toposes and be which an morphisms object. are geometric morphisms of(∞,1)(\infty,1)-toposes ff such that the inverse image f *f^* preserves small colimits and finite limits.

(2) (1) For The a restriction geometry of G X G X we to identify with LTop U(G) LTop(G) U is defined to be the slice(X∞/,U1) (\infty,1) X/U -category . which has as objectsGG-structures O:G→HO:G\to H on some (∞,1)(\infty,1)-topos HH and morphisms are those natural transformations f *f^* which (1.) are inverse images of geometric morphisms and (2.) whose naturality square is a pullback square in every admissible morphism.

(2) The restriction (O G,X) |U(O_{G,X})_{| U} of O G,XO_{G,X} to UU is defined to be composite
G→O G,XX→p *X/UG\stackrel{O_{G,X}}{\to}X\stackrel{p^*}{\to}X/U

where p *p^* is base change along p:U→*p:U\to *.
Definition (relative- ( and absolute spectrum)LTopLTop, LTop(G)LTop(G))

Let (1) p LTop:G→G 0 p:G\to LTop G_0 is defined to be a the morphism of geometries. Letp *:=(−∞ ) ,∘1p:LTop(G 0)→LTop(G) p^*:=(-)\circ (\infty,1) p:LTop(G_0)\to L Top(G) -category the which restriction objects functor. are(∞,1)(\infty,1)-toposes and which morphisms are geometric morphisms of (∞,1)(\infty,1)-toposes ff such that the inverse image f *f^* preserves small colimits and finite limits.

(1) (2) Then For there a is geometry an adjunctionGG we identify with LTop(G)LTop(G) the (∞,1)(\infty,1)-category which has as objects GG-structures O:G→HO:G\to H on some (∞,1)(\infty,1)-topos HH and morphisms are those natural transformations f *f^* which (1.) are inverse images of geometric morphisms and (2.) whose naturality square is a pullback square in every admissible morphism.

(Spec G,G 0⊣p *):LTop(G 0)→p *LTop(G)(Spec_{G,G_0}\dashv p^*):L Top(G_0)\stackrel{p^*}{\to}LTop(G)

where the left adjoint is called a relative spectrum functor.

(2) Let now G 0G_0 be the discrete geometry underlying GG. Then
Spec G:=Spec G,G 0∘ιSpec_G:= Spec_{G,G_0}\circ \iota

is called absolute spectrum functor; here ι:Ind(G op)↪LTop(G 0)\iota:Ind(G^{op})\hookrightarrow LTop(G_0) denotes the inclusion of the ind objects of GG.
Digression Definition (Example (relative- 2.3.8) and absolute spectrum)

Let H p:G→G 0 H p:G\to G_0 be a geometry, morphism let of geometries. Letfp *:U=(−)∘p:LTop(G 0)→ X LTop(G) f:U\to p^*:=(-)\circ X p:LTop(G_0)\to L Top(G) be the an restriction admissible functor. morphism inPro(H)Pro(H). Then

Spec HU→Spec HXSpec_H U\to Spec_H X

(1) Then there is an adjunction

is an étale morphism of absolute spectra.

(Spec G,G 0⊣p *):LTop(G 0)→p *LTop(G)(Spec_{G,G_0}\dashv p^*):L Top(G_0)\stackrel{p^*}{\to}LTop(G)

where the left adjoint is called a relative spectrum functor.

(2) Let now G 0G_0 be the discrete geometry underlying GG. Then
Spec G:=Spec G,G 0∘ιSpec_G:= Spec_{G,G_0}\circ \iota

is called absolute spectrum functor; here ι:Ind(G op)↪LTop(G 0)\iota:Ind(G^{op})\hookrightarrow LTop(G_0) denotes the inclusion of the ind objects of GG.
Proof Proposition (Theorem 2.2.12,(Spec H⊣Γ H)(\Spec_H\dashv \Gamma_H))

This Let follows from the following Proposition (Theorem 2.2.12).HH be a geometry, let XX be an object of Pro(G)Pro(G).

A (1) direct Then proof goes as follows: Letf:U→Xf:U\to X be admissible in Pro(H)Pro (H). Then Spec HSpec_H preserves finite limits, hence we can assume that ff arises from an admissible morphism f 0:U U→X 0f_0:U_U\to X_0 in HH.

Let Spec HX=:(χ,O χ)Spec_H X=:(\chi, O_\chi) such that O χ(X 0)O_\chi(X_0) has a canonical global section η:1 χ→O χ(X 0)\eta:1_\chi\to O_\chi(X_0). Let Y:=hfiber(η,O χ(f 0))Y:=hfiber(\eta, O_\chi(f_0)) and let

(Spec H⊣Γ H):LTop(H)→Γ HPro H(Spec_H\dashv \Gamma_H):LTop(H)\stackrel{\Gamma_H}{\to}Pro_H
(ϒ,O ϒ):=(χ/Y,O χ|Y)(\Upsilon,O_\Upsilon):=(\chi/Y, O_\chi |Y)

is an adjunction.

Then (2) there is a canonical global sectionη ′O Y \eta^\prime O_Y of factors asO ϒ(U 0)O_\Upsilon(U_0).

This means that η ′\eta^\prime exhibits (ϒ,O ϒ)(\Upsilon,O_\Upsilon) as an absolute spectrum UU such that Spec HfSpec_H f can be identified with the étale map (ϒ,O ϒ)→(χ,O χ)(\Upsilon, O_\Upsilon)\to (\chi,O_\chi).

H→jPro(H)→O Y¯YH\stackrel{j}{\to}Pro (H)\stackrel{\overline{O_Y}}{\to} Y

where jj denotes the Yoneda embedding and O Y¯\overline{O_Y} preserves small limits. And we have the identification of mapping spaces
Pro(H) op(X,Γ H(Y,O Y))≃Y(*,O Y¯(X))Pro(H)^{op}(X,\Gamma_H(Y,O_Y))\simeq Y(*, \overline{O_Y}(X))
Definition Digression (Example 2.3.8)

A Let G H G H -structured (∞,1)-topos be a geometry, let(f:U→X,O G,X) (X,O_{G,X}) f:U\to X is be called an admissible morphism inlocally representablePro(H)Pro(H) . (aka Then aGG-scheme) if

  • there exists a collection {U i∈X}\{U_i \in X\}
Spec HU→Spec HXSpec_H U\to Spec_H X

such is that an étale morphism of absolute spectra.

  • the {U i}\{U_i\} cover XX in that the canonical morphism ∐ iU i→*\coprod_i U_i \to {*} (with *{*} the terminal object of XX) is an effective epimorphism;

  • for every U iU_i there exists an equivalence

    (X/U i,O G,X| U i)≃Spec GA i (X/{U_i}, O_{G,X}|_{U_i}) \simeq Spec_{G} A_i

    of structured (∞,1)(\infty,1)-toposes for some A i∈Pro(G)A_i \in Pro(G) (in the (∞,1)-category of pro-objects in GG). In other words (X,O G,X)(X,O_{G,X}) is assumed to be locally equivalent to an absolute spectrum (aka affine scheme) of a pro object in GG.
Remark Proof

If This follows from the previous Proposition (Theorem 2.2.12).(C,O C)(C,O_C) is a structured topos and (C/U,(O C) |U)(C/U,(O_C)_{|U}) is an restriction thereof, then (C,O C)→(C/U,(O C) |U)(C,O_C)\to (C/U,(O_C)_{|U}) is an étale morphism of structured toposes.

A direct proof goes as follows: Let f:U→Xf:U\to X be admissible in Pro(H)Pro (H). Then Spec HSpec_H preserves finite limits, hence we can assume that ff arises from an admissible morphism f 0:U U→X 0f_0:U_U\to X_0 in HH.

Let Spec HX=:(χ,O χ)Spec_H X=:(\chi, O_\chi) such that O χ(X 0)O_\chi(X_0) has a canonical global section η:1 χ→O χ(X 0)\eta:1_\chi\to O_\chi(X_0). Let Y:=hfiber(η,O χ(f 0))Y:=hfiber(\eta, O_\chi(f_0)) and let
(ϒ,O ϒ):=(χ/Y,O χ|Y)(\Upsilon,O_\Upsilon):=(\chi/Y, O_\chi |Y)

Then there is a canonical global section η ′\eta^\prime of O ϒ(U 0)O_\Upsilon(U_0).

This means that η ′\eta^\prime exhibits (ϒ,O ϒ)(\Upsilon,O_\Upsilon) as an absolute spectrum UU such that Spec HfSpec_H f can be identified with the étale map (ϒ,O ϒ)→(χ,O χ)(\Upsilon, O_\Upsilon)\to (\chi,O_\chi).
Remark Definition

There A exists a H G H G -structure -structured (∞,1)-toposO(X,O G,X) O (X,O_{G,X}) on is called(H/X) fet(H/X)^{fet}locally representable such (aka that a((H/X) fet,O)((H/X)^{fet},O)GG-scheme ) is if a locally representableHH-structured (∞,1)(\infty,1)-topos.

  • there exists a collection {U i∈X}\{U_i \in X\}

such that

  • the {U i}\{U_i\} cover XX in that the canonical morphism ∐ iU i→*\coprod_i U_i \to {*} (with *{*} the terminal object of XX) is an effective epimorphism;

  • for every U iU_i there exists an equivalence

    (X/U i,O G,X| U i)≃Spec GA i (X/{U_i}, O_{G,X}|_{U_i}) \simeq Spec_{G} A_i

    of structured (∞,1)(\infty,1)-toposes for some A i∈Pro(G)A_i \in Pro(G) (in the (∞,1)-category of pro-objects in GG). In other words (X,O G,X)(X,O_{G,X}) is assumed to be locally equivalent to an absolute spectrum (aka affine scheme) of a pro object in GG.
Proof Remark

O:H→EO:H\to EIf has to satisfy(C,O C)(C,O_C) is a structured topos and (C/U,(O C) |U)(C/U,(O_C)_{|U}) is an restriction thereof, then (C,O C)→(C/U,(O C) |U)(C,O_C)\to (C/U,(O_C)_{|U}) is an étale morphism of structured toposes.

  • OO is left exact

  • OO satisfies codescent: For every collection of admissible (i.e. formally étale) morphisms {U i→X}\{U_i\to X\} in HH which generate a covering sieve on XX, the induced map ∐ iO(U i)→O(X)\coprod_i O(U_i)\to O(X) is an effective epimorphism in EE.

The terminal object in E:=(H/X) fetE:=(H/X)^{fet} is id Xid_X the identity on XX. The collection of all formally étale effective epimorphisms (in HH) with codomain XX covers XX. By HTT Remark 6.2.3.6. they cover id Xid_X in the slice.

Now we choose O:=r∘p *O:=r\circ p^* to be the composit of base change (this functor is exact) b *:H→H/Xb^*:H\to H/X along b:X→*b:X\to * followed by the coreflector (that we have a coreflector is shown (reference)) r:H/X→Er:H/X\to E (this functor is right adjoint and hence left exact). In total OO is left exact and since our cover consists only of formally étale morphisms rr and hence OO preserve the cover.

Now we describe the restriction of (E,O)(E,O) to an element UU of the cover:

Let U→id XU\to id_X be an element of the cover; i.e. a formally étale effective epimorphism U→XU\to X.

The restriction (E/U,O |U)(E/U,O_{| U}) of (E,O)(E,O) to UU is given by:

  • Objects of E/UE/U are cocones A → X ↘ ↙ U \array{A&\to &X\\\searrow &&\swarrow\\&U&} where A→XA\to X is formally étale. Morphisms are pyramids with four faces and tip UU.

  • The restriction of the HH-structure OO is given as follows:
H→OE→p *E/UH\stackrel{O}{\to}E\stackrel{p^*}{\to}E/U

where p *p^* is base change along p:U→*p:U\to *.

Now we show that (E,O)(E, O) is locally equivalent to an absolute spectrum:

Let H 0H_0 denote the discrete geometry (admissible morphisms are precisely all equivalences) with underlying category HH. Let h:H 0→Hh:H_0\to H be a morphism of geometries (i.e. hh preserves finite limits, maps admissible morphism to such, the image of an admissible cover is an admissible cover). Then there is an adjunction
(Spec H 0,H⊣p *):LTop(H 0)→Spec H 0,HLTop(H)(Spec_{H_0,H}\dashv p^*):LTop(H_0)\stackrel{Spec_{H_0,H}}{\to} LTop(H)

and the absolute spectrum Spec HSpec_H is defined to be the composit
Ind(H op)≃Pro(H) op≃Lex(H,∞Grpd)↪LTop(H 0)→Spec H 0,HLTop(H)Ind(H^{op})\simeq Pro(H)^{op}\simeq Lex(H, \infty Grpd)\hookrightarrow LTop(H_0)\stackrel{Spec_{H_0,H}}{\to}LTop(H)

(Pro objects in HH are cofiltered diagrams in HH or -equivalently - filtered diagrams in H opH^{op})
Remark

There exists a HH-structure OO on (H/X) fet(H/X)^{fet} such that ((H/X) fet,O)((H/X)^{fet},O) is a locally representable HH-structured (∞,1)(\infty,1)-topos.
Proof

O:H→EO:H\to E has to satisfy

  • OO is left exact

  • OO satisfies codescent: For every collection of admissible (i.e. formally étale) morphisms {U i→X}\{U_i\to X\} in HH which generate a covering sieve on XX, the induced map ∐ iO(U i)→O(X)\coprod_i O(U_i)\to O(X) is an effective epimorphism in EE.

The terminal object in E:=(H/X) fetE:=(H/X)^{fet} is id Xid_X the identity on XX. The collection of all formally étale effective epimorphisms (in HH) with codomain XX covers XX. By HTT Remark 6.2.3.6. they cover id Xid_X in the slice.

Now we choose O:=r∘p *O:=r\circ p^* to be the composit of base change (this functor is exact) b *:H→H/Xb^*:H\to H/X along b:X→*b:X\to * followed by the coreflector (that we have a coreflector is shown (reference)) r:H/X→Er:H/X\to E (this functor is right adjoint and hence left exact). In total OO is left exact and since our cover consists only of formally étale morphisms rr and hence OO preserve the cover.

Now we describe the restriction of (E,O)(E,O) to an element UU of the cover:

Let U→id XU\to id_X be an element of the cover; i.e. a formally étale effective epimorphism U→XU\to X.

The restriction (E/U,O |U)(E/U,O_{| U}) of (E,O)(E,O) to UU is given by:

  • Objects of E/UE/U are cocones A → X ↘ ↙ U \array{A&\to &X\\\searrow &&\swarrow\\&U&} where A→XA\to X is formally étale. Morphisms are pyramids with four faces and tip UU.

  • The restriction of the HH-structure OO is given as follows:

H→OE→p *E/UH\stackrel{O}{\to}E\stackrel{p^*}{\to}E/U

where p *p^* is base change along p:U→*p:U\to *.

Now we show that (E,O)(E, O) is locally equivalent to an absolute spectrum:

Let H 0H_0 denote the discrete geometry (admissible morphisms are precisely all equivalences) with underlying category HH. Let h:H 0→Hh:H_0\to H be a morphism of geometries (i.e. hh preserves finite limits, maps admissible morphism to such, the image of an admissible cover is an admissible cover). Then there is an adjunction

(Spec H 0,H⊣p *):LTop(H 0)→Spec H 0,HLTop(H)(Spec_{H_0,H}\dashv p^*):LTop(H_0)\stackrel{Spec_{H_0,H}}{\to} LTop(H)

and the absolute spectrum Spec HSpec_H is defined to be the composit

Ind(H op)≃Pro(H) op≃Lex(H,∞Grpd)↪LTop(H 0)→Spec H 0,HLTop(H)Ind(H^{op})\simeq Pro(H)^{op}\simeq Lex(H, \infty Grpd)\hookrightarrow LTop(H_0)\stackrel{Spec_{H_0,H}}{\to}LTop(H)

(Pro objects in HH are cofiltered diagrams in HH or -equivalently - filtered diagrams in H opH^{op})

Revision on December 17, 2012 at 18:03:24 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.

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